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 Mat. Sb. (N.S.), 1974, Volume 93(135), Number 2, Pages 218–253 (Mi msb2970)

Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra

G. I. Olshanskii

Abstract: In this paper we study the representations $\operatorname{Ind}(G,P,\pi)$ of the group $G=GL(n,D)$, where $D$ is a locally compact nondiscrete division algebra, that are induced by irreducible representations $\pi$ of an arbitrary parabolic subgroup $P\subset G$. If $D$ is totally disconnected, $\pi$ is assumed to be either supercuspidal (in the sense of Harish-Chandra; this is the same as absolutely cuspidal in the sense of Jacquet), or one-dimensional; we also allow combinations of these cases of a specific sort.
We give a construction of intertwining operators in this class of representations generalizing the construction of Schiffmann, Knapp and Stein. Using these intertwining operators, we prove that for the “principal series” representation $\operatorname{Ind}(G,P,\pi)$ to be contained in the “complementary series” the necessary formal condition of symmetry on $(P,\pi)$ turns out to also be sufficient. If $\pi$ is one-dimensional we estimate the width of the “critical interval”. Under certain conditions this estimate is best possible.
Bibliography: 28 titles.

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English version:
Mathematics of the USSR-Sbornik, 1974, 22:2, 217–255

Bibliographic databases:

UDC: 519.46
MSC: Primary 22E50, 12A70, 12A80; Secondary 22E45, 12B35

Citation: G. I. Olshanskii, “Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a locally compact division algebra”, Mat. Sb. (N.S.), 93(135):2 (1974), 218–253; Math. USSR-Sb., 22:2 (1974), 217–255

Citation in format AMSBIB
\Bibitem{Ols74} \by G.~I.~Olshanskii \paper Intertwining operators and complementary series in the class of representations induced from parabolic subgroups of the general linear group over a~locally compact division algebra \jour Mat. Sb. (N.S.) \yr 1974 \vol 93(135) \issue 2 \pages 218--253 \mathnet{http://mi.mathnet.ru/msb2970} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=499010} \zmath{https://zbmath.org/?q=an:0298.22016} \transl \jour Math. USSR-Sb. \yr 1974 \vol 22 \issue 2 \pages 217--255 \crossref{https://doi.org/10.1070/SM1974v022n02ABEH001692} 

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This publication is cited in the following articles:
1. J. H. Bernstein, A. V. Zelevinskii, “Induced representations of the group $GL(n)$ over a $p$-adic field”, Funct. Anal. Appl., 10:3 (1976), 225–227
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